Found problems: 6530
2019 Jozsef Wildt International Math Competition, W. 49
Let $a$, $b$, $c \in (0,+\infty)$ . Then the following inequality is true:$$\sqrt{(a+b)(b+c)}+\sqrt{(b+c)(c+a)}+\sqrt{(c+a)(a+b)}+a+b+c\leq \left(ab+bc+ca\right)\left(\frac{1}{\sqrt{ab}}+\frac{1}{\sqrt{bc}}+\frac{1}{\sqrt{ca}}\right)$$
2013 Silk Road, 3
Find all non-decreasing functions $ f\,:\,\mathbb{N}\to\mathbb{N} $, such that $f(f(m)f(n)+m)=f(mf(n))+f(m)$
2019 Jozsef Wildt International Math Competition, W. 16
If $f : [a, b] \to (0,\infty)$; $0 < a \leq b$; $f$ derivable; $f'$ continuous then:$$\int \limits_{a}^{b}\frac{f'(x)\sqrt{f(x)}}{f^3(x) + 1}\leq \tan^{-1}\left(\frac{f(b)-f(a)}{1 + f(a)f(b)}\right)$$
2021 Mediterranean Mathematics Olympiad, 4
Let $x_1,x_2,x_3,x_4,x_5$ ve non-negative real numbers, so that
$x_1\le4$ and
$x_1+x_2\le13$ and
$x_1+x_2+x_3\le29$ and
$x_1+x_2+x_3+x_4\le54$ and
$x_1+x_2+x_3+x_4+x_5\le90$.
Prove that $\sqrt{x_1}+\sqrt{x_2}+\sqrt{x_3}+\sqrt{x_4}+\sqrt{x_5}\le20$.
2021 Thailand TST, 2
Suppose that $a,b,c,d$ are positive real numbers satisfying $(a+c)(b+d)=ac+bd$. Find the smallest possible value of
$$\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}.$$
[i]Israel[/i]
2006 China Northern MO, 8
Given a sequence $\{ a_{n}\}$ such that $a_{n+1}=a_{n}+\frac{1}{2006}a_{n}^{2}$ , $n \in N$, $a_{0}=\frac{1}{2}$.
Prove that $1-\frac{1}{2008}< a_{2006}< 1$.
2013 Miklós Schweitzer, 12
There are ${n}$ tokens in a pack. Some of them (at least one, but not all) are white and the rest are black. All tokens are extracted randomly from the pack, one by one, without putting them back. Let ${X_i}$ be the ratio of white tokens in the pack before the ${i^{\text{th}}}$ extraction and let
\[ \displaystyle T =\max \{ |X_i-X_j| : 1 \leq i \leq j \leq n\}.\]
Prove that ${\Bbb{E}(T) \leq H(\Bbb{E}(X_1))},$ where ${H(x)=-x\ln x -(1-x)\ln(1-x)}.$
[i]Proposed by Tamás Móri[/i]
2018 Belarusian National Olympiad, 10.2
Determine, whether there exist a function $f$ defined on the set of all positive real numbers and taking positive values such that $f(x+y)\geqslant yf(x)+f(f(x))$ for all positive x and y?
2010 China Team Selection Test, 2
Find all positive real numbers $\lambda$ such that for all integers $n\geq 2$ and all positive real numbers $a_1,a_2,\cdots,a_n$ with $a_1+a_2+\cdots+a_n=n$, the following inequality holds:
$\sum_{i=1}^n\frac{1}{a_i}-\lambda\prod_{i=1}^{n}\frac{1}{a_i}\leq n-\lambda$.
2012 Indonesia TST, 1
Let $a,b,c \in \mathbb{C}$ such that $a|bc| + b|ca| + c|ab| = 0$. Prove that $|(a-b)(b-c)(c-a)| \ge 3\sqrt{3}|abc|$.
2009 ELMO Problems, 3
Let $a,b,c$ be nonnegative real numbers. Prove that \[ a(a - b)(a - 2b) + b(b - c)(b - 2c) + c(c - a)(c - 2a) \geq 0.\][i]Wenyu Cao[/i]
2011 Sharygin Geometry Olympiad, 4
Segments $AA'$, $BB'$, and $CC'$ are the bisectrices of triangle $ABC$. It is known that these lines are also the bisectrices of triangle $A'B'C'$. Is it true that triangle $ABC$ is regular?
2002 VJIMC, Problem 3
Positive numbers $x_1,\ldots,x_n$ satisfy
$$\frac1{1+x_1}+\frac1{1+x_2}+\ldots+\frac1{1+x_n}=1.$$Prove that
$$\sqrt{x_1}+\sqrt{x_2}+\ldots+\sqrt{x_n}\ge(n-1)\left(\frac1{\sqrt{x_1}}+\frac1{\sqrt{x_2}}+\ldots+\frac1{\sqrt{x_n}}\right).$$
2021 Irish Math Olympiad, 9
Suppose the real numbers $a, A, b, B$ satisfy the inequalities: $$|A - 3a| \le 1 - a\,\,\, , \,\,\, |B -3b| \le 1 - b$$, and $a, b$ are positive. Prove that $$\left|\frac{AB}{3}- 3ab\right | - 3ab \le 1 - ab.$$
2013 Romania National Olympiad, 4
a) Consider\[f\text{:}\left[ \text{0,}\infty \right)\to \left[ \text{0,}\infty \right)\] a differentiable and convex function .Show that $f\left( x \right)\le x$, for every $x\ge 0$, than ${f}'\left( x \right)\le 1$ ,for every $x\ge 0$
b) Determine \[f\text{:}\left[ \text{0,}\infty \right)\to \left[ \text{0,}\infty \right)\] differentiable and convex functions which have the property that $f\left( 0 \right)=0\,$, and ${f}'\left( x \right)f\left( f\left( x \right) \right)=x$, for every $x\ge 0$
2008 Tournament Of Towns, 3
In his triangle $ABC$ Serge made some measurements and informed Ilias about the lengths of median $AD$ and side $AC$. Based on these data Ilias proved the assertion: angle $CAB$ is obtuse, while angle $DAB$ is acute. Determine a ratio $AD/AC$ and prove Ilias' assertion (for any triangle with such a ratio).
2022 China Girls Math Olympiad, 8
Let $x_1, x_2, \ldots, x_{11}$ be nonnegative reals such that their sum is $1$. For $i = 1,2, \ldots, 11$, let
\[ y_i = \begin{cases} x_{i} + x_{i + 1}, & i \, \, \textup{odd} ,\\ x_{i} + x_{i + 1} + x_{i + 2}, & i \, \, \textup{even} ,\end{cases} \]
where $x_{12} = x_{1}$. And let $F (x_1, x_2, \ldots, x_{11}) = y_1 y_2 \ldots y_{11}$.
Prove that $x_6 < x_8$ when $F$ achieves its maximum.
2005 QEDMO 1st, 12 (U2)
For any three positive real numbers $a$, $b$, $c$, prove the inequality \[\frac{\left(b+c\right)^{2}}{a^{2}+bc}+\frac{\left(c+a\right)^{2}}{b^{2}+ca}+\frac{\left(a+b\right)^{2}}{c^{2}+ab}\geq 6.\] Darij
1975 USAMO, 1
(a) Prove that \[ [5x]\plus{}[5y] \ge [3x\plus{}y] \plus{} [3y\plus{}x],\] where $ x,y \ge 0$ and $ [u]$ denotes the greatest integer $ \le u$ (e.g., $ [\sqrt{2}]\equal{}1$).
(b) Using (a) or otherwise, prove that \[ \frac{(5m)!(5n)!}{m!n!(3m\plus{}n)!(3n\plus{}m)!}\] is integral for all positive integral $ m$ and $ n$.
1969 IMO Shortlist, 15
$(CZS 4)$ Let $K_1,\cdots , K_n$ be nonnegative integers. Prove that $K_1!K_2!\cdots K_n! \ge \left[\frac{K}{n}\right]!^n$, where $K = K_1 + \cdots + K_n$
2015 Thailand TSTST, 3
Let $a, b, c$ be positive real numbers. Prove that
$$\frac {3(ab + bc + ca)}{2(a^2b^2+b^2c^2+c^2a^2)}\leq \frac1{a^2 + bc} + \frac1{b^2 + ca} + \frac1{c^2 + ab}\leq\frac{a+b+c}{2abc}.$$
2017 Saudi Arabia JBMO TST, 1
Let $a,b,c>0$ and $abc=1$ . Prove that $$ \sqrt{2(1+a^2)(1+b^2)(1+c^2)}\ge 1+a+b+c.$$
2020 Iran MO (2nd Round), P2
let $x,y,z$ be positive reals , such that $x+y+z=1399$ find the
$$\max( [x]y + [y]z + [z]x ) $$
( $[a]$ is the biggest integer not exceeding $a$)
2015 Postal Coaching, Problem 1
Find all positive integer $n$ such that
$$\frac{\sin{n\theta}}{\sin{\theta}} - \frac{\cos{n\theta}}{\cos{\theta}} = n-1$$
holds for all $\theta$ which are not integral multiples of $\frac{\pi}{2}$
2019 Thailand TSTST, 1
Let $\{x_i\}^{\infty}_{i=1}$ and $\{y_i\}^{\infty}_{i=1}$ be sequences of real numbers such that $x_1=y_1=\sqrt{3}$, $$x_{n+1}=x_n+\sqrt{1+x_n^2}\quad\text{and}\quad y_{n+1}=\frac{y_n}{1+\sqrt{1+y_n^2}}$$
for all $n\geq 1$. Prove that $2<x_ny_n<3$ for all $n>1$.